Linear actions of groups + algebras: finite-group representations, Lie-algebra reps, Hecke algebras, modular reps, geometric representation theory, quantum groups.
representation-theory
Character theory (Frobenius)
Frobenius 1896 first character-theory of finite groups. Character χ_V(g) = tr ρ(g). Orthogonality relations → number of irreducibles =…
Induced representation + Frobenius reciprocity
Ind_H^G V = G-rep induced from H-rep V via tensor product k[G] ⊗_{k[H]} V. Frobenius-reciprocity Hom_G(Ind W, V) = Hom_H(W, Res V). Mackey…
Highest weight rep (cross-listed L0 lie-theory)
Cross-listed L0 lie-theory representation-theory-lie-framework. Verma + simple highest-weight modules. Weyl character formula. Foundation…
Iwahori-Hecke algebra
Iwahori 1964 / Hecke earlier. Deformation of group-algebra of Weyl-group. Type-A: H_n(q) = quotient of braid-group-algebra. Foundation of…
Modular representation theory (Brauer)
Brauer 1935-50s: rep-theory in characteristic p dividing |G|. p-blocks + defect-groups. Brauer character. Modular Atlas. Distinct from…
Category of representations (irreducible decomposition)
Maschke 1898: char(k) ∤ |G| ⇒ k[G] semisimple ⇒ every rep decomposes into irreducibles. Failure for modular case. Schur's lemma: Hom(V,W)…
Tannakian formalism
Tannaka-Krein 1938 / Saavedra 1972 / Deligne 1990: reconstruct algebraic-group from its category of representations. Tensor + duality +…
Geometric representation theory (flag variety)
Reps realised as cohomology of flag-varieties: Borel-Weil-Bott theorem H^*(G/B, L_λ). Beilinson-Bernstein localisation. Springer…
Quantum groups (Drinfeld-Jimbo)
Drinfeld-Jimbo 1985 q-deformation U_q(g) of universal-enveloping algebra. Hopf-algebra structure. Reps form braided-tensor-category →…
Automorphic representations (Langlands)
Langlands 1967- conjectural correspondence: automorphic-reps of GL_n(A_F) ↔ n-dim Galois-reps. Foundation of modern number-theory unifying…
Vertex operator algebras (Borcherds)
Frenkel-Lepowsky-Meurman 1988 / Borcherds 1986 (Fields 1998). Algebra structure on monstrous-moonshine module V^♮. Foundation for 2D-CFT +…
Modular tensor category (CFT)
Verlinde 1988 / Moore-Seiberg 1989 / Reshetikhin-Turaev 1990. Modular-tensor-category encodes 2D-CFT data; gives 3-manifold invariants.…
Schur orthogonality relations
Schur 1905: irreducible representations of finite group G satisfy orthogonality across characters; basis of character-theory +…
Frobenius reciprocity
Frobenius 1898 / Brauer: Hom_G(Ind^G_H V, W) iso Hom_H(V, Res^G_H W); foundation of induced-representation theory + intertwiner calculation.
Peter-Weyl theorem
Peter-Weyl 1927: regular representation of compact Lie group decomposes as direct sum of irreducibles; matrix-coefficients form…
Borel-Weil theorem (flag variety)
Borel-Weil 1954: highest-weight irreducible representations of compact connected Lie group realized on holomorphic-section spaces of line…
Langlands program (foundational)
Langlands 1967 conjectures: deep links between number-theory L-functions + automorphic representations + Galois groups; partial proofs…
Kazhdan-Lusztig polynomials
Kazhdan-Lusztig 1979: combinatorial polynomials parametrize irreducible representations of Hecke algebras + Weyl groups; decisive role in…
Frobenius (1896)
G Frobenius 1896 character-theory finite-groups; modern modern foundational text + Burnside 1911 + 2024 modular-rep Lusztig +…
Schur (1905)
I Schur 1905 + 1924 Schur-Weyl duality GL(n)-S_n; modern modern foundational text + 2024 quantum-groups + categorification q-Schur algebra.
Weyl (1939)
H Weyl 1939 'Classical Groups'; modern modern foundational text + Borel-Weil-Bott + 2024 geometric-Langlands GLGT representations.
Harish-Chandra (1956)
Harish-Chandra 1956 + Langlands 1967 admissible-rep semisimple-Lie-groups; modern modern foundational + 2024 Langlands-program…
Kazhdan-Lusztig (1979)
D Kazhdan-G Lusztig 1979 Hecke-KL-polynomials; modern modern foundational text + 2024 categorification + Soergel-bimodules computation.
Beilinson-Bernstein (1981)
A Beilinson-J Bernstein 1981 D-module-rep + Brylinski-Kashiwara 1981 KL-conjecture; modern modern foundational + 2024 geometric-Langlands.
Schur-Weyl duality (GL(n) / S_k commutant on (C^n)^{tensor k})
Schur-Weyl duality (Schur 1901 dissertation; Weyl 1939 *Classical Groups*) states that on the k-fold tensor power V^{tensor k} of an…